Bihamiltonian approach to the gauge models: closed string model
The closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. It is shown that bihamiltonity is origin of two types of the T-duality...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2001 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/79425 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Bihamiltonian approach to the gauge models: closed string model / V.D. Gershun // Вопросы атомной науки и техники. — 2001. — № 6. — С. 65-70. — Бібліогр.: 9 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859632823678271488 |
|---|---|
| author | Gershun, V.D. |
| author_facet | Gershun, V.D. |
| citation_txt | Bihamiltonian approach to the gauge models: closed string model / V.D. Gershun // Вопросы атомной науки и техники. — 2001. — № 6. — С. 65-70. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. It is shown that bihamiltonity is origin of two types of the T-duality of the closed string models. The dual nonlocal Poisson brackets, depending of the background fields and of their derivatives, are obtained. The integrability condition is formulated as the compatibility of the bihamoltonity condition and the Jacobi identity of the dual Poisson bracket. It is shown, that the dual brackets and dual hamiltonians can be obtained from the canonical (PB) and from the initial hamiltonian by imposing of the second kind constraints on the initial dynamical system, on the closed string model in the constant background fields, as example. The closed string model in the constant background fields is considered without constraints, with the second kind constraints and with first kind constraints as the B-chiral string. The two particles discrete closed string model is considered as two relativistic particle system to show the difference between the Gupta-Bleuler method of the quantization with the first kind constraints and the quantization of the Dirac bracket with the second kind constraints.
|
| first_indexed | 2025-12-07T13:12:31Z |
| format | Article |
| fulltext |
BIHAMILTONIAN APPROACH TO THE GAUGE MODELS:
CLOSED STRING MODEL
V.D. Gershun
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The closed string model in the background gravity field and the antisymmetric B-field is considered as the
bihamiltonian system in assumption that string model is the integrable model for particular kind of the background
fields. It is shown that bihamiltonity is origin of two types of the T-duality of the closed string models. The dual
nonlocal Poisson brackets, depending of the background fields and of their derivatives, are obtained. The
integrability condition is formulated as the compatibility of the bihamoltonity condition and the Jacobi identity of
the dual Poisson bracket. It is shown, that the dual brackets and dual hamiltonians can be obtained from the
canonical (PB) and from the initial hamiltonian by imposing of the second kind constraints on the initial dynamical
system, on the closed string model in the constant background fields, as example. The closed string model in the
constant background fields is considered without constraints, with the second kind constraints and with first kind
constraints as the B-chiral string. The two particles discrete closed string model is considered as two relativistic
particle system to show the difference between the Gupta-Bleuler method of the quantization with the first kind
constraints and the quantization of the Dirac bracket with the second kind constraints.
PACS:11.17.-w 11.30.-j 11.30.Na 11.30.Lm
1. INTRODUCTION
The bihamiltonian approach [1-3] to the integrable
systems was initiated by Magri [4] for the investigation
of the integrability of the KdV equation. A finite
dimensional dynamical system with 2N degrees of
freedom xa, a=1...2N is integrable, if it is described by
the set of the n integrals of motion F1,...,Fn in involution
under some Poisson bracket (PB)
{Fi,Fk}PB=0. (1)
The dynamical system is completely solvable, if n=N.
Any of the integral of motion (or any linear combination
of them) can be considered as the hamiltonian Hk = Fk
.The bihamiltonity condition has following form:
NN
aa
a
a HxHx
dt
dxx },{...},{ 11 ==== . (2)
The hierarchy of new (PB) is arosed in this
connection:
{ , }1, { , }2,..,{ , }N . (3)
The hierarchy of new dynamical systems is arosed
under the new time coordinates tk.
1},{ +
+
= kn
a
kn
a
Hx
dt
dx
. (4)
The new equations of motion are described the new
dynamical systems, which are dual to the original
system, with the dual set of the integrals of motion. The
dual set of the integrals of motion can be obtained from
original one by the mirror transformations and by the
contraction of the integrals of motion algebra. The
contraction of the integral of motion algebra means, that
the dynamical system is belong to the orbits of
corresponding generators and is describe the invariant
subspace. The set of the commuting integrals of motion
belongs to Cartan subalgebra of this algebra.
Consequently, duality is property of the integrable
models. KdV equation is one of the most interesting
examples of the infinite dimensional integrable
mechanical systems with soliton solutions.
We are considered the dynamical systems with
constraints. In this case, first kind constraints are the
generators of the gauge transformations and they are
integrals of motion. First kind constrains Fk(xa)≈0,
k=1,2... form the algebra of constraints under some
(PB).
{ , }F F C Fi k PB ik
l
l= ≈ 0 . (5)
The structure functions Cl
ik may be functions of the
phase space coordinates in general case. The second
kind constraints fk(xa) ≈ 0 are the representations of the
first kind constraints algebra. The second kind
constraints is defined by the condition
{fi, fk }=Cik ≠0. (6)
The reversible matrix Cik is not constraint and it is
function of phase space coordinates also. The second
kind constraints take part in deformation of the { , }PB to
the Dirac bracket { , }D. As rule, such deformation leads
to the nonlinear and to the nonlocal brackets. First kind
constraints are imposed upon the vector states under the
quantization: Fk|Ψ〉 =0. The same spectrum of the
excitations and the wave functions are obtained under
the Gupta-Bleuer method of the quantization. One-half
of the second kind constraints can be considered as first
kind constraints and they must be imposed upon vector
states in Gupta-Bleuer method of quantization. The (PB)
is not deformed to the Dirac bracket in this connection.
The bihamiltonity condition leads to the dual (PB),
which are nonlinear and nonlocal brackets as rule.
We suppose, that the dual brackets can be obtained
from the initial canonical bracket under the imposition
of the second kind constraints. We make this conclusion
from the consideration of two dynamical models, closed
string model in the constant background fields and two
particles discrete closed string model, as examples. The
Gupta-Bleyer method of the quantization may be more
preferable in some case, if the dual bracket is nonlocal.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, 65-70. 65
We have applied [5,6] bihamiltonity approach to the
investigation of the integrability of the closed string
model in the arbitrary background gravity field and
antisymmetric B-field. The bihamiltonity condition and
the Jacobi identities for the dual brackets have
considered as the integrability condition for a closed
string model. They led to some restrictions on the
background fields. The local dual (PB) of the similar
type have considered [7] in the application to the
hamiltonian hydrodynamical models. The (PB) of the
hydrodynamical type for the phase coordinate functions
ui(x,t) is defined by formula
).())((
)())(()}(),({
yxuxub
yxxugyuxu
l
x
ik
l
x
ikki
−+
−=
δ
δ∂
(7)
There gik(u), bik
l(u) are the arbitrary functions of the
phase space coordinates and u ux x= ∂ . The Jacobi
identity is satisfied under the following conditions:
1. Tensor gik is symmetric tensor and it is define some
metric on the phase space.
2. bik
l(u)=-gij Γk
ij and connection Γk
ij is consistent to
metric gik and it has zero curvature and zero torsion.
Therefore, there are such local coordinates, that gik=
const, bik
l=0. This (PB) was used for description of the
hamiltonian system of the hydrodynamical type. That is
systems with functionals of the hydrodynamical type.
The density of this functionals does not depend of the
derivatives uk
x, uk
xx,... and hamiltonian is functional of
the hydrodynamical type also. In opposite this models,
the functionals of the closed string model is depended of
the derivatives of the string coordinates. As result, we
need to introduce additional nonlocal term with the step
function )(2)( 1 yxyx x −=− − δ∂ε , which is the
origin of the difficulty of the Jacobi identity proof.
The plan of the paper is following. In the second
section we are considered closed string model in the
arbitrary background gravity field and antisymmetric B-
field as the bihamiltonian system. We suppose, that this
model is integrable model for some configurations of
the background fields. The bihamiltonity condition and
the Jacobi identities for the dual (PB) must be result to
the integrability condition, which is restricted the
possible configurations of the background fields. The
well-known examples of the integrable gravity models
with the gravity metric tensor, which is depended of one
or of two variables only.
In this paper we are assumed the metric dependence
of the arbitrary number of the variables for generality
and we did not analyzed the particular cases of the
metric dependence. In the third section we are
considered three examples of the closed string model in
the constant background fields: without constraints,
with the second kind constraints and the B-chiral string
with the first kind constraints. In the four section we are
considered two particles discrete closed string model to
show the difference between the (PB) structure under
the Gupta-Bleuer method of the quantization and under
the quantization of the system with the second kind
constraints.
2. CLOSED STRING IN THE BACKGROUND
FIELDS
The closed string in the background gravity field and
the antisymmetric B-field is described by first kind
constraints
,0,0)(
2
1
])(][)()[(
2
1
2
1
≈′=≈′′+
′−′−=
a
a
ba
ab
d
bdb
c
aca
ab
xphxxxg
xxBpxxBpxgh αα
(8)
where a,b =0,1,...D-1, xa(σ), pa(σ) are the periodical
functions on σ with the period on π , α -arbitrary
parameter. The original (PB) are the symplectic (PB)
.0},{},{
)()}(),({
11
1
==
′−=′
ba
ba
a
bb
a
ppxx
px σσδδσσ
(9)
The hamiltonian equations of motion of the closed
string, in the arbitrary background gravity field and
antisymmetric B-field under the hamiltonian
H h d1 1
0
= ∫ σ
π
and (PB) {,}1, are
[ ]x g p B xa ab
b bc
c= − ′α ,
.][
][
2
1
)(
)(
2
1
][
2
2
2
cb
ec
de
adac
b
cb
eb
de
bdbca
c
bdc
adb
c
bdc
bdacba
bc
b
db
cd
acabc
bc
aba
xxBgBg
x
xxBgBg
x
pxgB
x
pxgB
x
pp
x
g
xBgBgpgBp
′′−+
+′′−−
−′+
+′−−
′′−+′=
α
∂
∂
α
∂
∂
∂
∂α
∂
∂α
∂
∂
αα
(10)
The dual (PB) are obtained from the bihamiltonity
condition
,},{},{
},{},{
2
0
21
0
1
2
0
21
0
1
σσ
σσ
ππ
ππ
′=′=
′=′=
∫∫
∫∫
dhpdhpp
dhxdhxx
aaa
aaa
(11)
and they have the following form:
{ ( ), ( )}
[[ ( ) ( )] ( )
[ ( ) ( )] ( )
[ ( ) ( )] ( )]
A B
A
x
B
xa b
ab ab
ab ab
ab ab
σ σ
∂
∂
∂
∂
ω σ ω σ ε σ σ
σ σ
∂
∂ σ
σ σ
σ σ δ σ σ
′ =
+ ′ ′ − +
+ ′
′
′ − +
+ ′ ′ − +
2
Φ Φ
Ω Ω
(12)
+σ−σ ′δσ ′Ω+σΩ
+σ−σ ′
σ ′∂
∂σ ′Φ+σΦ
+σ−σ ′εσ ′ω+σω
∂
∂
∂
∂
)]()]()([
)()]()([
)()]()([[
abab
abab
abab
ba p
B
p
A
66
+σ−σ ′εσ ′ω+σω
∂
∂
∂
∂+
∂
∂
∂
∂ )()]()(][[ a
b
a
ba
bb
a x
B
p
A
p
B
x
A
)].()]()(][[
)]()]()([
σ−σ ′δσ ′Ω+σΩ
∂
∂
∂
∂−
∂
∂
∂
∂
+σ−σ ′
σ ′∂
∂σ ′Φ+σΦ
a
b
a
ba
bb
a
a
b
a
b
x
B
p
A
p
B
x
A
The arbitrary functions A, B, ω, Φ, Ω are the functions
of the xa (σ), pa(σ ). The functions ωab, ωab, Φab, Φab are
the symmetric functions on a, b and Ωab, Ωab are the
antisymmetric functions to satisfy the condition
{A, B}2 = -{B, A}2. The equations of motion under the
hamiltonian H2 and (PB) { , }2 are
,)(
)(
)(]
[222
242
0
c
b
b
a
cb
b
a
c
cb
b
ac
b
b
ac
cb
b
ac
c
c
b
ac
ba
bb
abba
b
ba
b
b
ab
b
abba
b
a
xp
p
x
x
pp
p
x
x
pp
p
px
x
xdpxx
ppxx
′′Φ+′Φ−
−′′Φ+′Φ+
−′′+′
+′′+′Ω−′Ω+′′Φ
−′′Φ++−=
∫
∂
∂
∂
∂
∂
∂
∂
∂
σσε
∂
∂ ω
∂
∂ ω
ωσ
ωω
π
(13)
.)(
)(
)()
(224
222
2
0
cb
b
c
ab
b
c
a
c
b
b
acb
b
ac
cba
bc
dc
b
ba
cd
b
abb
b
ab
b
ab
b
a
b
ab
b
ab
b
aba
pp
p
x
x
xp
p
x
x
pp
x
ppx
xx
xdppp
xxxp
′′Φ+′Φ+
+′′Φ+′Φ−
−−′′−′−
−′′+′Ω+′′Φ++
+′Ω+′′Φ−−=
∫
∂
∂
∂
∂
∂
∂
∂
∂
σσε
∂
∂ ω
∂∂
ω∂
ωσω
ω
π
(14)
The bihamiltonity condition (11) is led to the two
constraints
,
)(
)(
)(]
[222
242
0
c
bc
ab
b
ab
c
b
b
a
cb
b
a
c
cb
b
ac
b
b
ac
cb
b
ac
c
c
b
ac
ba
bb
abba
b
ba
b
b
ab
b
abba
b
xBgpg
xp
p
x
x
pp
p
x
x
pp
p
px
x
xdpxx
ppx
′−=
=′′Φ+′Φ−
−′′Φ+′Φ+
−′′+′
+′′+′Ω−′Ω+′′Φ
−′′Φ++−
∫
α
∂
∂
∂
∂
∂
∂
∂
∂
σσε
∂
∂ ω
∂
∂ ω
ωσ
ωω
π
(15)
=′′Φ+′Φ+
+′′Φ+′Φ−
−−′′−′−
−′′+′Ω+′′Φ++
+′Ω+′′Φ−−
∫
cb
b
c
ab
b
c
a
c
b
b
acb
b
ac
cba
bc
dc
b
ba
cd
b
abb
b
ab
b
ab
b
a
b
ab
b
ab
b
ab
pp
p
x
x
xp
p
x
x
pp
x
ppx
xx
xdppp
xxx
)(
)(
)()
(224
222
2
0
∂
∂
∂
∂
∂
∂
∂
∂
σσε
∂
∂ ω
∂∂
ω∂
ωσω
ω
π
(16)
.][
][
2
1
)(
)(
2
1
][
2
2
2
cb
ec
de
adac
b
cb
eb
de
bdbca
c
bdc
adb
c
bdc
bdacba
bc
b
db
cd
acabc
bc
ab
xxBgBg
x
xxBgBg
x
pxgB
x
pxgB
x
pp
x
g
xBgBgpgB
′′−+
+′′−−
−′+
+′−−
−′′−+′=
α
∂
∂
α
∂
∂
∂
∂α
∂
∂α
∂
∂
αα
In really, there is the list of the constraints depending on
the possible choice of the unknown functions ω, Φ, Ω.
In the general case, there are as the first kind constraints
as the second kind constraints too. Also, it is possible to
solve the constraints equations as the equations for the
definition of the functions ω, Φ, Ω. We are considered
last possibility and we obtained the following consistent
solution of the bihamiltonity condition:
],[
2
1
,
2
,0,0,0
2
2
db
cd
acabab
cb
ac
a
bdcba
cd
ab
ababa
b
abab
BgBg
p
x
gCpp
xx
gC
Cg
α
∂
∂
ω
∂∂
∂
ω
ω
−−=Φ
−==
==Φ=Ω=Φ
(17)
.0,,2
)2(2
1,)(
)(
2
1
==−=Ω
+
=Ω−Ω−
−′Φ−Φ=Ω
c
ab
abc
c
ab
cb
aca
b
cb
c
a
a
c
b
c
b
ac
a
bc
ab
p
ngx
x
gBg
n
Cp
xx
x
xx
∂
∂ ω
∂
∂α
∂
∂
∂
∂
∂
∂
∂
∂
The metric tensor gab(x) is the homogeneous function of
xa order n and C is arbitrary constant. In the difference
of the (PB) of the hydrodynamical type, we are needed
to introduce the separate (PB) for the coordinates of the
Minkowski space and for the momenta because, the
gravity field is not depend of the momenta. Although,
this difference is vanished under the such constraint as
f(xa,pa)≈0. One can see, that the main term in the (PB)
with the metric tensor is the term with the step function
ε(σ−σ′). The functions ωb
a, Ωab are proportional to the
connection and to the torsion. The function ωab is
proportional to the curvature and the product of the
connections. The dual (PB) for the phase space
coordinates are:
{ ( ), ( )} [ ( ) ( )] ( )x xa b ab abσ σ ω σ ω σ ε σ σ′ = + ′ ′ −2 ,
).()]()([
)()]()([)}(),({
)],()]()([
)()]()([
)()]()([)}(),({
2
2
σσδσσ
σσεσωσωσσ
σσδσσ
σσ
σ∂
∂σσ
σσεσωσωσσ
−′′Ω+Ω+
+−′′+=′
−′′Ω+Ω
+−′
′
′Φ+Φ
+−′′+=′
a
b
a
b
a
b
a
bb
a
abab
abab
ababba
px
pp
(18)
The functions ωab, ωab, Φab, Ωab, ωa
b, Ωa
b are defined in
(17). It is rather easy to prove the Jacobi identities for
the local part of the dual (PB) { , }2. It does not
understand, how to prove the Jacobi identities for the
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
67
nonlocal part of it. The principal term of the Jacobi
identities with the step functions only is the term with
the structure function ωab(x):
σ
∂
σ∂σσ
∂
σ∂
σσεσσεσ
σ
∂
σ∂σσ
∂
σ∂
′
′
−+′
′
+−′′−′′+
−′′+
([)()]()([)([
)()()]](
)([)()]()([)([
dc
d
ab
dcda
d
cb
db
db
d
ac
dcdc
d
ab
g
x
ggg
x
g
g
g
x
ggg
x
g
.0)()()]](
)([)()]()([)([
)()()]](
=′′−′′′−+
′′′′
−′+′′′′
+′−′′′−′′+
σσεσσεσ
σ
∂
σ∂σσ
∂
σ∂
σσεσσεσ
da
da
d
cb
dbdb
d
ac
dc
g
g
x
ggg
x
g
g
(19)
It is possible to reduce this condition to the unique
equation
.0)()()]](
)([)()]()([)([
=−′′−′′+
−′′+
σσεσσεσ
σ
∂
σ∂σσ
∂
σ∂
db
db
d
ac
dcdc
d
ab
g
g
x
ggg
x
g
(20)
One of the possible ways of the solution of this problem
is the consideration of the metric tensor on the phase
space to count the contribution of the structure functions
Φ and Ω to this expression. The second possible way is
the consideration of the second kind constraints from
the list of the constraints (15,16), instead of the solution
(17). It is necessary to introduce this constraints to the
initial model with the hamiltonian H1, the (PB) { , }1
and to obtain the bihamiltonity condition in this case. As
we will see later on the some examples, the second kind
constraints of the f(xa,pa)≈0 type can lead to the
nonlocal Dirac bracket with the step function from the
one side, and they can introduce the dependence of the
metric tensor of the momenta on the solutions of this
constraints from the other side. At present, these
problems are under the consideration.
3. CONSTANT BACKGROUND FIELDS
The bihamiltonity condition (15,16) is reduced to the
following constraints on the phase space:
.][
22
4224
,0,2
2242
2 b
db
cd
acab
c
bc
abb
b
ab
b
a
b
b
a
b
ab
b
ab
b
ab
abc
bc
ab
b
abba
b
ba
bb
ab
b
abba
b
xBgBg
pgBpp
pxxx
xBgpgx
xppx
′′−+
+′=′Ω+′′Φ
++′Ω+′′Φ−−
=Ω′−=′Ω
+′′Φ−′′Φ++−
α
α
ωω
α
ωω
(21)
There is the unique solution without constraints
.2
,2,
4
1
2
db
cd
acabab
ca
bccb
aca
b
abab
BgBg
gBBgg
α
ααω
−=Φ−
=−=Ω=
(22)
The rest structure functions are equal zero. In this
section we are supplemented the bihamiltonity condition
(11) by the mirror transformations of the integrals of
motion.
{ , } { , }x x H x Ha a a= = ± ±1 1 2 2 . (23)
The dual (PB) are:
).(][
)}(),({
),()}(),({
),(
2
1)}(),({
2
2
2
2
σσδ
σ∂
∂α
σσ
σσδασσ
σσεσσ
−′
′
−=
=′
−′=′
−′±=′
±
±
±
db
cd
acab
ba
cb
ac
b
a
abba
BgBg
pp
Bgpx
gxx
(24)
The dual dynamical system
2112 },{},{ ±=±= HxHxx aaa (25)
is the left(right) chiral string :
aa
aa ppxx ′=′±= , . (26)
In the terms of the Virasoro operators
∫
∫
−=
+=
π
σ
π
σ
σ
π
σ
π
0
21
0
21
}(
4
1
,}(
4
1
dehhL
dehhL
ik
k
ik
k
(27)
the first kind constraints form the Vir⊕ Vir algebra
under the (PB) { , }1:
mnmn LmniLL +−−= )(},{ 1 ,
mnmn LmniLL +−−= )(},{ 1 , (28)
.0},{ 1 =mn LL
The dual set of the integrals of motion is obtained from
initial it by the mirror transformations
H H L L L L1 2 0 0 0 0→ → ± → →, , ,τ σ (29)
and by the contraction of the first kind constraints
algebra Ln=0, or, L nn = ≠0 0, .
3.1. SECOND KIND CONSTRAINTS
Another way to obtain the dual brackets is the
imposition of the second kind constraints on the initial
dynamical system, by such manner, that Fi=Fk for i≠ k, i,
k =1,2,... on the constraints surface f(xa,pa)=0.
Let us consider the closed string model with Bab=0 for
simplicity:
.
,
2
1
2
1
2
1
a
a
ba
abba
ab
xph
xxgppgh
′=
′′+=
(30)
The constraints f x p p g xa a ab
b( ) ( , ) ,− = − ′ ≈ 0
or 0),()( ≈′+=+ b
abaa xgppxf (do not simultaneously)
are the second kind constraints:
68
).(2
)()}(),({ )(
1
)()(
σσδ
σ∂
∂
σσσσ
−′
′
±=
=′−=′ ±±±
ab
abba
g
Cff
(31)
The inverse matrix (C(±)
ab )-1 has following form:
).(
4
1)()( σσεσσ −′±=′−± abab gC (32)
There is only one set of the constraints, because
consistency conditions
{ ( ), } ( ) ,...
{ ( ), } ( )
( ) ( )
( )( ) ( )( )
f H f
f H fn n
± ±
± + ±
= ′ ≈
= ≈
σ σ
σ σ
1 1
1 1
1
0
0
(33)
are not produce the new sets of constraints. By using the
standard definition of the Dirac bracket, we are obtained
following Dirac brackets for the phase space
coordinates:
{ ( ), ( )} ( ),
{ ( ), ( )} ( ),
{ ( ), ( )} ( ).
x x g
p p g
x p
a b
D
ab
a b D ab
a
b D b
a
σ σ ε σ σ
σ σ
∂
∂ σ
δ σ σ
σ σ δ δ σ σ
′ = ± ′ −
′ =
′
′ −
′ = ′ −
1
4
1
2
1
2
(34)
The equations of motion under the hamiltonians H1 =h1,
H2 =h2 and Dirac bracket
a
b
abDaDaa
a
b
ab
D
a
D
aa
pxgHpHpp
xpgHxHxx
′±=′′===
′±====
},{},{
,},{},{
21
21
(35)
are coincide on the constraints surface. The dual
brackets { , }(±2) are coincide with the Dirac brackets
also. The contraction of the algebra of the first kind
constraints means that the integrals of motion H1=H2 are
coincide on the constraints surface too.
3.2 B-CHIRAL STRING
Let us consider the following constraint from the list (24)
ϕ βa a ab
bp B x= + ′ ≈ 0 (36)
The consistency condition
{ , } ( )
[ ( ) ]
ϕ α β ϕ
α β
a ab
bc
c
ab ac
cd
db
b
H B g
g B g B x
1 1
2
= + ′ +
+ − + ′′
(37)
shows that under the additional condition on the B-field
g B g Bab ac
cd
db= +( )α β 2 , (38)
the constraints ϕa ≈0 are first kind constraints. The
motion equations are:
( ) ,
( ) , .
x g B x
p B g p x x
a ab
bc
c
a ab bc c
a a
= − + ′
= − + ′ = ′′
α β
α β
(39)
This model is the bihamiltonian model under (PB) (26)
also. The B-chiral string model is dual to the chiral
model also.
4. TWO PARTICLES DISCRETE STRING
In this section we are considered two particles
discrete closed string model, as two relativistic particles
model, to show the difference between the Dirac
brackets quantization and the Gupta-Bleuer quantization
methods.
Two and three pieces discrete string in Gupta-Bleuer
method of quantization was considered in the paper [8]
and it is described by the following constraints:
h p q r= + +
1
4
2 2
0
2 2( ) ,ω (40)
f pq f qr1 30 0= ≈ = ≈, ,
f pr f q r2 4
2
0
2 20
1
4= ≈ = −, ω .
This model is the two relativistic particles system with
the oscillator interaction. The constraints (40) are two
particles discrete analog of the Virasoro constraints and
p, q, r are the collective variables pa=p2
a+p1
a, qa=p2
a-p1
a,
r=x2
a-x1
a.
Under the hamiltonian H=h and the canonical (PB)
{ra,qb}=ηab, the constraints fi are the second kind
constraints:
{ , } ,
{ , } .
f f p
f f q r
1 2
2
3 4
2
0
2 2
2 0
4 0
= ≠
= + ≠ω
(41)
The string coordinates are satisfied to following Dirac
brackets:
).
4
4(2},{
},,{4},{,},{
22
0
2
2
0
2
022
0
2
rq
rrqq
p
ppqr
rrqq
rq
qrqrrr
bababa
abDba
baDba
baab
Dba
ω
ωη
ω
ω
+
+−−=
=
+
−
=
(42)
In the terms of the amplitudes aa
(+),aa of the equations of
motion solutions
r a e a e
q i a e a e
a a
i
a
i
a a
i
a
i
= +
= −
+ −
+ −
ω τ ω τ
ω τ ω τω
0 0
0 02 0
( )
( )
,
( )
(43)
they have the form:
{ , }
( )
,
{ , } { , } ,
{ , } ( )
( ) ( )
( )
( ) ( )
( )
r r
i a a a a
a a
q q r r
r q
p p
p
a a a a
a a
a b D
b a a b
k k
a b D a b D
a b D ab
a b a b b a
k k
=
−
=
= − −
+
+ +
+
+ +
+
2
2
2
2
0
0
2
ω
ω
η
(44)
This Dirac brackets is possible to solve in the terms of
the variables a, aa
(+).
{ , } ( ).( )
( )
( )a a
i p p
p
ia a
a aa b D ab
a b a b
k k
+
+
+= − −
2 20
2
0ω
η
ω
(45)
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
69
The hamiltonian and the linear combinations of the
constraints f1, f4 have following form:
.,,,
,4
4
1
)()(
43
)(
21
)(2
0
2
+++
+
====
+=
kkkkkkkk
kk
aafaafapfapf
aapH ω
(46)
Under the quantization [ , ]→{ , }D, ak, ak
(+) → operators
ak, ak
(+) the commutation relation is
[ , ] ( )
( ) .
( )
( ) ( )
a a
i p p
p
a a a a
k l kl
k l
i i k l
+
+ − +
=
−
− +
+
2
1
2
0
2
0
1
ω
η
ω
(47)
Last term of the commutation relation has transposed
Lorentz indices k,l. The wave function
〉Ψ=〉Ψ +++ 0|)(...)(| ...
)()()(
2121
paaap
nn kkkkkkn (48)
of the physical states must to be the own function of the
hamiltonian H on the constraints surface. Let us
consider the two particles excited state, for example.
〉Ψ−=〉Ψ ++ 0|)()4
4
1()(| )()(2
0
2
2 2121 kkkk aapppH ω +
terms, which are proportional to the expressions:
p p p a a pk k k k k k k l l k k1 1 2 1 2 1 2 1 2
Ψ Ψ Ψ( ), ( ), ( )( ) ( )η + + . (49)
Consequently, we must to impose additional conditions
on the wave function pkΨkl, ηklΨkl=0 to satisfy the
request about the own function. Last term ai
(+) ai
(+) Ψ kl is
vanished on the constraints surface. In contrast to the
nonlinear and to the nonlocal Dirac bracket (45), we
have the canonical (PB) and two first kind constraints
pkak≈0 , akak≈0 in the Gupta-Bleuer method of the
quantization. The first kind constraints H, f1, f3 are
imposed on the vector states and they are led to the
following equations on the wave function:
.0)(,0)(
,0)()16(
......
...
2
0
2
21
=Ψ=Ψ
=Ψ−
ppp
pp
nn
n
kklklkkkk
kkk
η
ω
(50)
5. RELATIVISTIC PARTICLE IN THE
CONSTANT ELECTROMAGNETIC FIELD
The relativistic particle in the constant background
electromagnetic field is described by the Hamiltonian
H p i B x ma ab b= + +
1
2
2 2[( ) ]β . (51)
The electromagnetic field is Aa(x)=-2Fabxb=-Babxb.
The simplest constraint from (21) is
ϕa=pa+iαBabxb≈0. (52)
The consistency condition
{ϕa,H}=iα(α+β)Babϕb (53)
shows that there is a unique set of constraints if α+β=0.
They are the second class constraints:
{ϕa,ϕb}=2iαBab.
There is the following algebra of the phase space
coordinates under the Dirac bracket
.
2
},{
,
2
1},{,)(
2
},{ 1
abDba
abDbaabDba
Bipp
pxBixx
α
η
α
−=
=−= −
(54)
The motion equations under the Dirac bracket
, x i B x p B xa ab b a ab b+ = + =2 0 2 0α α (55)
have the solution x e xa
i B
ab b( ) ( ) ( ).τ α τ= − 2 0
The quantization of the Dirac bracket results in the
following commutation relations:
.
2
],[
,
2
],[,)(
2
1],[ 1
abba
abbaabba
ipx
BppBxx
η
α
α
=
== −
(56)
ACKNOWLEDGMENTS
The author would like to thanks J. Lukierski for the
kind hospitality in the Wroclaw University and
A.I. Pashnev for useful discussions.
REFERENCES
1. L.D. Fadeev and L.A. Takhtajan. Hamiltonian
methods in theory of solitons. Berlin: "Springer", 1987,
528 p.
2. J.A. Mitropolski, N.N. Bogolubov (Jr.), A.K. Pri-
karpatski, V.G. Samoylenko, Integrable dynamical
systems:spectral and differential geometric aspects.
Kiev: "Naukova Dumka", 1987, 296 p. (in Russian).
3. S. Okubo, A. Das,The integrability condition for
dynamical systems // Phys. Lett. 1988, B 209, p. 311-
314.
4. F.A. Magri. A geometrical approach to the nonlinear
solvable equations // Lect. Notes Phys. 1980, v. 120,
p. 233-263.
5. V.D. Gershun. Bihamiltonity as origin of the T-
duality of the closed string model // Nucl. Phys. B
(Proc. Suppl.). 2001, v 102&103, p. 71-76, hep-
th/0103097.
6. V.D. Gerhun. T-duality of string models in the
bachground gravity and antisymmetric fields. A
Collection of Papers: Gravitaton, Cosmology and
Relativistic Astrophysics. Kharkov: “Kharkov
National University”, 2001, p. 69-74.
7. Background gravity and antisymmetric fields.
Gravitation, cosmology and relativistic
astrophysycs, Kharkov: "KNU", 2001, p. 69-74.
8. B.A. Dubrovin, S.P. Novikov. Hydrodynamics of
the weak deformed soliton solutions // Sov. Uspekhi
Math. 1989, v. 44, p. 29-98.
70
9. V.D. Gershun, A.I. Pashnev. Relativistic system of
interacting points as a discrete string // Teor. Mat.
Fiz. 1987, v. 73, p. 294-301.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
71
V.D. Gershun
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
1. INTRODUCTION
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79425 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:12:31Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Gershun, V.D. 2015-04-01T19:30:22Z 2015-04-01T19:30:22Z 2001 Bihamiltonian approach to the gauge models: closed string model / V.D. Gershun // Вопросы атомной науки и техники. — 2001. — № 6. — С. 65-70. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS:11.17.-w 11.30.-j 11.30.Na 11.30.Lm https://nasplib.isofts.kiev.ua/handle/123456789/79425 The closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. It is shown that bihamiltonity is origin of two types of the T-duality of the closed string models. The dual nonlocal Poisson brackets, depending of the background fields and of their derivatives, are obtained. The integrability condition is formulated as the compatibility of the bihamoltonity condition and the Jacobi identity of the dual Poisson bracket. It is shown, that the dual brackets and dual hamiltonians can be obtained from the canonical (PB) and from the initial hamiltonian by imposing of the second kind constraints on the initial dynamical system, on the closed string model in the constant background fields, as example. The closed string model in the constant background fields is considered without constraints, with the second kind constraints and with first kind constraints as the B-chiral string. The two particles discrete closed string model is considered as two relativistic particle system to show the difference between the Gupta-Bleuler method of the quantization with the first kind constraints and the quantization of the Dirac bracket with the second kind constraints. The author would like to thanks J. Lukierski for the kind hospitality in the Wroclaw University and A.I. Pashnev for useful discussions. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Bihamiltonian approach to the gauge models: closed string model Бигамильтонов подход к калибровочным моделям: модель замкнутой струны Article published earlier |
| spellingShingle | Bihamiltonian approach to the gauge models: closed string model Gershun, V.D. Quantum field theory |
| title | Bihamiltonian approach to the gauge models: closed string model |
| title_alt | Бигамильтонов подход к калибровочным моделям: модель замкнутой струны |
| title_full | Bihamiltonian approach to the gauge models: closed string model |
| title_fullStr | Bihamiltonian approach to the gauge models: closed string model |
| title_full_unstemmed | Bihamiltonian approach to the gauge models: closed string model |
| title_short | Bihamiltonian approach to the gauge models: closed string model |
| title_sort | bihamiltonian approach to the gauge models: closed string model |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79425 |
| work_keys_str_mv | AT gershunvd bihamiltonianapproachtothegaugemodelsclosedstringmodel AT gershunvd bigamilʹtonovpodhodkkalibrovočnymmodelâmmodelʹzamknutoistruny |